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On a Neumann-type series for modified Bessel functions of the first kind

Authors: L. Deleaval and N. Demni
Journal: Proc. Amer. Math. Soc. 146 (2018), 2149-2161
MSC (2010): Primary 33C45, 33C52, 33C65
Published electronically: December 28, 2017
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Abstract: In this paper, we are interested in a Neumann-type series for modified Bessel functions of the first kind which arises in the study of Dunkl operators associated with dihedral groups and as an instance of the Laguerre semigroup constructed by Ben Said-Kobayashi-Orsted. We first revisit the particular case corresponding to the group of square-preserving symmetries for which we give two new and different proofs other than the existing ones. The first proof uses the expansion of powers in a Neumann series of Bessel functions, while the second one is based on a quadratic transformation for the Gauss hypergeometric function and opens the way to derive further expressions when the orders of the underlying dihedral groups are powers of two. More generally, we give another proof of De Bie et al.'s formula expressing this series as a $ \Phi _2$-Horn confluent hypergeometric function. In the course of the proof, we shed light on the occurrence of multiple angles in their formula through elementary symmetric functions and get a new representation of Gegenbauer polynomials.

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Additional Information

L. Deleaval
Affiliation: Laboratoire d’Analyse et de Mathématiques appliquées Université Paris-Est Marne-la-Vallée 77454 Marne-la-Vallé Cedex 2 France

N. Demni
Affiliation: IRMAR, Université de Rennes 1 Campus de Beaulieu 35042 Rennes cedex France

Keywords: Modified Bessel functions, Gegenbauer polynomials, generalized Bessel function, dihedral groups, elementary symmetric functions.
Received by editor(s): March 31, 2017
Received by editor(s) in revised form: July 22, 2017, and July 29, 2017
Published electronically: December 28, 2017
Communicated by: Yuan Xu
Article copyright: © Copyright 2017 American Mathematical Society

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